IAEA, TECDOC, Chapter 6

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1 IAEA, TECDOC, Chapter 6 Pecker, Johnson, Jeremić Draft Writeup (in progress, total up to 50 pages) version: 27. September, 2016, 22:31

2 Contents 6 Methods and models for SSI analysis Basic steps for SSI analysis Direct methods (Jeremic) Linear and Nonlinear Discrete Methods Finite Element Method Equilibrium Equations Finite Element Equations Finite Elements Finite Difference Method Finite Difference Solution Technique Nonlinear discrete methods Inelasticity, Elasto-Plasticity Material Models for Dynamic Modeling Nonlinear Dynamics Solution Techniques Sub-structuring methods (Pecker and Johnson) Sub-Structuring Methods, Principles and Numerical Implementation (Pecker) Soil Structure Interaction CLASSI: A Linear Continuum Mechanics Approach (Johnson) Discrete methods Foundation input motion SSI computational models (Jeremic and Pecker) Introduction Soil/Rock Linear and Nonlinear Modelling Effective and Total Stress Analysis Dry Soil Partially Saturated Soil Saturated Soil Drained and Undrained Modeling Drained Analysis Undrained Analysis Linear and Nonlinear Elastic Models Equivalent Linear Elastic Models Elastic-Plastic Models A Note on Constitutive Level and Global Level Equilibrium Structural models, linear and nonlinear: shells, plates, walls, beams, trusses, solids Contact Modeling Contact Modeling Formulation

3 Elastic behavior Plastic model Geometry description Structures with a base isolation system Base Isolation Systems Base Dissipator Systems Foundation models Shallow and Embedded Slab Foundations Piles and Shaft Foundations Deeply Embedded Foundations Foundation Flexibility and Base Isolator/Dissipator Systems Small Modular Reactors (SMRs) Buoyancy Modeling Dynamic Buoyant Stress/Force Modeling Domain Boundaries Seismic Load Input The Domain Reduction Method A Note on Free Field Input Motions for DRM Liquefaction and Cyclic Mobility Modeling Introduction Liquefaction Modeling Details and Discussion Structure-Soil-Structure Interaction Simplified models Simplified Models Simplified, Discrete Soil and Structural Models P-Y and T-Z Springs Simplified, Continuum Soil Models Linear Elastic Stiffness Reduction (G/G max ) and Damping Curve Models General guidance on soil structure interaction modelling and analysis Model Development Model Verification Probabilistic response analysis (Jeremic and Johnson) Introduction Probabilistic Response Analysis Monte Carlo Random Vibration Theory Stochastic Finite Element Method

4 Chapter 6 Methods and models for SSI analysis Jeremić et al. ( ) (50 Pages) (Pecker & Jeremic as Chapter leads) 6.1 Basic steps for SSI analysis To identify candidate SSI models, model parameters, and analysis procedures, assess: The purposes of the SSI analysis (design and/or assessment): Seismic response of structure for design or evaluation (forces, moments, stresses or deformations, such as story drift, number of cycles of response) Input to the seismic design, qualification, evaluation of subsystems supported in the structure (in-structure response spectra ISRS; relative displacements, number of cycles) Base-mat response for base-mat design Soil pressures for embedded wall designs The characteristics of the subject ground motion (seismic input motion): Amplitude (excitation level) and frequency content (low vs. high frequency) Low frequency content (2 Hz to 10 Hz) affects structure and subsystem design/capacity; high frequency content (> 20 Hz) only affects operation of mechanical/electrical equipment and components; Incoherence of ground motion; 4

5 Are ground motions 3D? Are vertical motions coming from P or S (surface) waves. If from S and surface waves, we have full 3D motions. What to do about it (model?) Refer to Chapters 3, 4, and 5 for free-field ground motion and seismic input discussions The characteristics of the site: Idealized site profile is applicable (Section ) Linear or equivalent linear soil material model applicable (visco- elastic model parameters assigned) Nonlinear (inelastic, elastic-plastic) material model necessary? Non-idealized site profile necessary? Sensitivity studies to be performed to clarify model requirements for site characteristics (complex site stratigraphy, inelastic modeling, etc.)? The structure characteristics: Expected behavior of structure (linear or nonlinear); Based on initial linear model of the structure, perform preliminary seismic response analyses (response spectrum analyses) to determine stress levels in structure elements; If significant cracking or deformations possible (occur) such that portions of the structure behave nonlinearly, refine model either approximately introducing cracked properties or model portions of the structure with nonlinear elements; For expected structure behavior, assign material damping values; The foundation characteristics: Effective stiffness is rigid due to base-mat stiffness and added stiffness due to structure being anchored to base-mat, e.g., honey-combed shear walls anchored to base-mat; Effective stiffness is flexible, e.g. if additional stiffening by the structure is not enough to claim rigid; or for strip footings; The end result is to identify the important elements of SSI to be considered in the analysis of the subject structure: 5

6 Seismic input as defined in Chapters 4 and 5; Equivalent linear vs. nonlinear (inelastic, elastic-plastic) soil behavior; equivalent linear substructure approach to SSI acceptable; Linear, equivalent linear, or nonlinear (inelastic) structure behavior; equivalent linear implements approximate stiffness degradation for structures; linear/equivalent linear - substructure approach to SSI acceptable; Foundation to be modeled as behaving rigidly (e.g., first stage of multi-stage analysis) or flexibly; Select SSI model and analysis procedure. For the SSI model and analysis to be implemented, confirm existence of Verification for all models, elements, etc. Validation for all (as many as possible) models, etc. Determine the application domain for all models, elements, etc. (Application domain is discussed in some detail in section on Verification and Validation, in Chapter 8). Before initial results are available, make estimates of what type of behavior you expect to see (accomplished in steps above and confirmed herein). In (both) cases, if results are similar or not similar to your pre-analysis expectations do the following investigations: investigate alternative parameters, in order to understand sensitivity of results to parameter variations, investigate alternative models (with different degree of fidelity, simplifications, etc.), in order to understand sensitivity of results to (simplifying) modeling assumptions Modeling sequence should be: Linear elastic, model components first then slowly complete the model: soil only, static loads (point, self-weight, etc.); dynamic loads (point loads, etc.); free field ground motions (see chapter 4 and 5) 6

7 components of structural model only (for example containment only, internal structure only, etc.), and then full structural model (just the structure, no soil), static loads (self weight in three directions to verify model and load paths, point loads to verify model and load paths); then dynamic loads (point loads, and seismic loads) complete structure and foundation, (apply same load scenarios as above) complete structure, foundation, soil system, (apply same load scenarios as above) Equivalent linear modeling, and observe changes in response, to determine possible plastification effects. It is very important to note that it is still an elastic analysis, with reduced (equivalent) linear stiffness. Reduction in secant stiffness really steams from plastification, although plastification is not explicitly modeled, hence an idea can be obtained of possible effects of reduction of stiffness. One has to be very careful with observing these effects, and focus more on verification of model (for example wave propagation through softer soil, frequencies will be damped, etc...). Nonlinear/inelastic modeling, slowly introduce nonlinearities to test models, convergence and stability, in all the components as above. Investigate sensitivities for both linear elastic and nonlinear/inelastic simulations! 6.2 Direct methods (Jeremic) Linear and Nonlinear Discrete Methods Linear and nonlinear mechanics of solids and structures relies on equilibrium of external and internal forces/stresses. such equilibrium can be expressed as σ ij,j = f i ρü i (6.1) where σ ij,j is a small deformation (Cauchy) stress tensor, f i are external (body (fi B) and surface (f i S) ) forces, ρ is material density and ü i are accelerations. Inertial forces ρü i follow from d Alembert s principle (D Alembert, 1758). The above equation forms a basis for both Finite Element Method (FEM) and Finite Difference Method (FDM). Above equation can be pre-multiplied with virtual displacements δu i and then integrated by parts to obtain the weak form, as further elaborated below in section This equation can also be directly solved using finite differences, as noted in section

8 It is important to note that equation 6.1 is usually not satisfied in either FEM or FDM. Rather is is satisfied in an approximate fashion, with a smaller or large deviation, depending on type of FEM or FDM used. Finite Element Method Equilibrium Equations Development of finite element equations is efficiently done by using principle of virtual displacements. This principle states that the equilibrium of the body requires that for any compatible, small virtual displacements, which satisfy displacement boundary conditions imposed onto the body, the total internal virtual work is equal to the total external virtual work. Finite Element Equations After some manipulations (Zienkiewicz and Taylor, 1991a,b), we can write the finite element equations as: M P Q ū P + C P Q ū P + K P Q ū P = F Q P, Q = 1, 2,..., (#ofdof s)n (6.2) where M P Q is a mass matrix, C P Q is a damping matrix, K P Q is a stiffness matrix and F Q is a force vector. Damping matrix C P Q cannot be directly developed from a formulation for a single phase solid or structure. In other words, viscous damping is a results of interaction of fluid and solid/structure and is not part of this formulation (Argyris and Mlejnek, 1991). Viscous damping can, however, be added through viscoelastic constitutive material models and through Rayleigh damping, or a more general, Caughey damping. Viscous damping can also be added through viscoelastic constitutive material models. In general Caughey damping is defined as (Semblat, 1997): m 1 C = [M] a j ([M] 1 [K]) j (6.3) j=0 where the order used depends on number of modes to be considered for damping in the problem. The second order Caughey damping, is also known as a Rayleigh damping, with j = 1 in Equation (6.3). In reality, damping matrix (more precisely, damping resulting from viscous effects) results from an interaction of soils and/or structures with surrounding fluids (Argyris and Mlejnek, 1991). For porous solid with pore space filled with fluid, a direct derivation of damping matrix is possible (Jeremić et al., 2008). Stiffness matrix K P Q can be linear (elastic) or nonlinear, elastic-plastic. 8

9 Finite element analysis comprises a discretization of a solid and/or structure into an assemblage of discrete finite elements. Finite elements are connected at nodal points. It is very important to note that the finite element method is an approximate method. Generalized displacement solutions at nodes are approximate solutions. A number of factors controls the quality of such approximate solutions. For example it can be shown (Zienkiewicz and Taylor, 1991a,b; Hughes, 1987; Argyris and Mlejnek, 1991) that an increase in a number of nodes, finite elements (refinement of discretization) and a reduction of increments (loads steps or time step size) will lead to a more accurate solution. However, this refinement in mesh discretization and reduction of step size, will lead to longer run times. A fine balance needs to be achieved between accuracy of the solution and run time. This is where verification procedures (described in some details in section 8.) become essential. Verification procedures provide us magnitudes of errors that we can expect from our finite element (approximate) solutions. Results from verification procedures should thus be used to decide appropriate discretization (in space (mesh) and load/time) to achieve desired accuracy in solution. Finite Elements There exist different types of finite elements. They can be broadly classified into: Solid elements (3D brick, 2D quads etc.) Structural elements (truss, beam, plate, shell, etc.) Special Elements (contacts, etc.) Solid finite elements usually feature displacement unknowns in nodes, 3 displacements for 3D elements, and 2 unknowns displacements for 2D elements. The most commonly used 3D solid finite elements are bricks, that can have 8, 20, and 27 nodes. In 3D, tetrahedral elements (4 and 10 nodes) are also popular due to their ability to be meshed into any volume, while solid brick elements sometimes can have problems with meshing. In 2D most common are quads, with 4, 8 or 9 nodes (Zienkiewicz and Taylor, 1991a,b; Bathe, 1996a). Triangular elements are also popular (3, 6 and 10 nodes), due to the same reason, that is triangles can be meshed in any plane shape, unlike quads. Two dimensional finite elements can approximate plane stress, plane strain or axisymmetric continuum. It is important to note that 3 node triangular elements feature constant strain field, and thus lead to discontinuous strains, and possibility of mesh locking. Solid finite elements are also used to model coupled problems where porous solid (soil skeleton) is coupled with pore fluid (water), as described by Zienkiewicz and Shiomi (1984); Zienkiewicz et al. (1990, 1999). These elements and the underlying formulation will be described in some detail in section

10 Structural finite elements use integrated section stress to develop section generalized forces (normal, transversal and moments). Truss elements can have 2 or 3 nodes. Beams usually have 2 nodes, although 3 node beam elements are also used (Bathe, 1996a). Most beam elements are based on a Euler-Bernoulli beam theory, which means that they do not take into account shear deformation, and thus should only be used for slender beams, where the ratio of beam length to (larger) beam cross section dimension is more than 10 (some authors lower this number to 5) (Bathe, 1996a). For beams that are not slender, Timoshenko beam element is recommended (Challamel, 2006), as it explicitly takes into account shear deformation. Plate, wall and shell elements are usually quads or triangles. Plate finite elements model plate bending without taking into account forces in the plane of the plane plate. Main unknowns are transversal displacement and two bending (in plane) moments. In plane forcing and deformation is modeled using wall elements that are very similar to plane stress 2D elements noted above. In plane nodal rotations are usually not taken into account. If possible, it is beneficial to include rotational (drilling) degree of freedom (Bergan and Felippa, 1985), so that wall elements has three degrees of freedom per node (two in plane displacements and out of plane rotation). Shell element is obtained by combining plate bending and wall elements. Special elements are used for modeling contacts, base isolation and dissipation devices and other special structural and contact mechanics components of an NPP soil-structure system (Wriggers, 2002). Finite Difference Method Finite different methods (FDM) operate directly on dynamic equilibrium equation 6.1, when it is converted into dynamic equations of motion. The FDM represents differentials in a discrete form. It is best used for elasto-dynamics problems where stiffness remains constant. In addition, it works best for simple geometries (Semblat and Pecker, 2009), as finite difference method requires special treatment boundary conditions, even for straight boundaries that are aligned with coordinate axes. Finite Difference Solution Technique The FDM solves dynamic equations of motion directly to obtain displacements or velocities or accelerations, depending on the problem formulation. Within the context of the elasto-dynamic equations, on which FDM is based, elastic-plastic calculations are performed by changes to the stiffness matrix, in each step of the time domain solution. 10

11 Nonlinear discrete methods Nonlinear problems can be separated into (Felippa, 1993; Crisfield, 1991, 1997; Bathe, 1996a) Geometric nonlinear problems, involving smooth nonlinearities (large deformations, large strains), and Material nonlinear problems, involving rough nonlinearities (elasto-plasticity, damage) Main interest in modeling of soil structure interaction is with material nonlinear problems. Geometric nonlinear problems are involve large deformations and large strains and are not of much interest here. It should be noted that sometimes contact problems where gaping occurs (opening and closing or gaps) are called geometric nonlinear problems. They are not geometric nonlinear problems for cases of interest here, namely, gap opening and closing between foundations. Problems where gap opens and closes are material nonlinear problems where material stiffness (and internal forces) vary between very small values (zeros in most formulations) when the gap is opened, and large forces when the gap is closed. Material nonlinear problems can be modeled using Linear elastic models, where linear elastic stiffness is the initial stiffness or the equivalent elastic stiffness (Kramer, 1996; Semblat and Pecker, 2009; Lade, 1988; Lade and Kim, 1995). Initial stiffness uses highest elastic stiffness of a soil material for modeling. It is usually used for modeling small amplitude vibrations. These models can be used for 3D modeling. Equivalent elastic models use secant stiffness for the average high estimated strain (typically 65% of maximum strain) achieved in a given layer of soil. Eventual modeling is linear elastic, with stiffness reduced from initial to approximate secant. These models should really be only used for 1D modeling. Nonlinear 1D models, that comprise variants of hyperbolic models (described in section 3.2), utilize a predefined stress-strain response in 1D (usually shear stress τ versus shear strain γ) to produce stress for a given strain. There are other nonlinear elastic models also, that define stiffness change as a function of stress and/or strain changes (Janbu, 1963; Duncan and Chang, 1970; Hardin, 1978; Lade and Nelson, 1987; Lade, 1988) 11

12 These models can successfully model 1D monotonic behavior of soil in some cases. However, these models cannot be used in 3D. In addition, special algorithmic measures (tricks) must be used to make these models work with cyclic loads. Elastic-Plastic material modeling can be quite successfully used for frys frys both monotonic, and cyclic loading conditions (Manzari and Dafalias, 1997; Taiebat and Dafalias, 2008; Papadimitriou et al., 2001; Dafalias et al., 2006; Lade, 1990; Pestana and Whittle, 1995). Elastic plastic modeling can also be used for limit analysis (de Borst and Vermeer, 1984). Inelasticity, Elasto-Plasticity Inelastic, elastic-plastic modeling relies on incremental theory of elasto-plasticity to solve elastic-plastic constitutive equations, with appropriate/chosen material model. Most solutions are strain driven, while there exist techniques to exert stress and mixed control (Bardet and Choucair, 1991). There are two levels of nonlinear/inelastic modeling when elasto-plasticity is employed: Constitutive level, where nonlinear constitutive equations with appropriate material models are solved for stress and stiffness (tangent or consistent) given strain increment Global, finite element level, where nonlinear dynamic finite element equations are solved for given dynamic loads and current (elastic-plastic) stiffness (tangent or consistent). Material Models for Dynamic Modeling At the constitutive level, general 3D strain increments (incremental strain tensor, or in other words, increments in all six independent components of strain, normal (σ xx, σ yy, and σ yy ) and shear (σ xy, σ yz, and σ zx )) is driving the nonlinear constitutive solution. Proper elastic-plastic material models must be chosen to obtain results. Elastic-plastic material models, consist of four main components: Elasticity, that governs the elastic response, before material yields. Yield function, a function in stress and internal variables (shear strength, friction angle, back-stress, etc.) space, that separates elastic region from the elastic-plastic region. Plastic flow directions, that provide directions of plastic strain, once material plastifies. Magnitude of plastic strain is obtained from the solution of constitutive equations. 12

13 Hardening/softening rules, that control evolution of yield surface and plastic flow direction, during plastic deformation. There are four main types of hardening/softening rules, that can be combined between each other (for example isotropic and kinematic hardening models can be combined): Perfect plastic material behavior, where yield function and plastic flow directions do not change during plastic deformation. There is no internal variable for this type of hardening/softening. Isotropic hardening/softening material behavior, where yield function and plastic flow directions change isotropically (proportionally). This type of hardening/softening is only good for monotonic loading and should not be used for cyclic loading. Internal variables are of scalar type, for example friction angle, shear strength, maximum isotropic confinement, etc. Kinematic hardening where yield function and plastic flow direction either translate (works well for metals and total stress analysis of undrained, soft clays), or rotate (works well for soils, concrete, rock and other pressure sensitive materials). This type of hardening (usually it is only used for hardening, there is no softening) is good for cyclic loading. Internal variables are the back stress. Distortional hardening where yield function and plastic flow direction can have a general change in stress and internal variable space. This type of hardening/softening is the most general case and contains all the previous hardening/softening cases, however it is rarely used, as it requires a large number of tests. Dynamic modeling, where stresses and strain cyclically change requires models that feature kinematic hardening. In case of pressure sensitive materials, like soil, concrete and rock, rotational kinematic hardening is used. For materials that do not have pressure sensitivity (metals, and saturated clays when modeled with a total stress approach (as opposed to effective stress approach (Jeremić et al., 2008)), translational kinematic hardening is used. There exist a number of models developed recently that can produce satisfactory modeling of dynamic response of geomaterials (Dafalias and Manzari, 2004; Taiebat and Dafalias, 2008; Dafalias et al., 2006; Mróz et al., 1979; Mroz and Norris, 1982; Prevost and Popescu, 1996). Of particular importance is availability of calibration tests, and addressing the issue of uncertainty and sensitivity of material response to changes in parameters. 13

14 It is also important to address the issue of spatial variability and uncertainty in material parameters for soils, as the ensuing response can also be quite uncertain. The issue of spatial variability and uncertainty in material modeling will be addressed in more detail in section will be addressed in section 6.5. Nonlinear Dynamics Solution Techniques On the global, finite element level, finite element equations are solved using time marching algorithms. Most often used are Newmark algorithm (Newmark, 1959) and Hilber-Hughes-Taylor (HHT) α algorithm (Hilber et al., 1977). Other algorithms (Wilson θ, l Hermite, etc.) also do exist Argyris and Mlejnek (1991); Hughes (1987); Bathe and Wilson (1976), however they are used less frequently. Both Newmark and HHT algorithm allow for numerical damping to be included in order to damp out higher frequencies that are introduced artificially into FEM models by discretization of continua into discrete finite elements. Solution to the dynamic equations of motion can be done by either enforcing or not enforcing convergence to equilibrium. Enforcing the equilibrium usually requires use of Newton or quasi Newton methods to satisfy equilibrium within some tolerance. This results in a (much) longer running times, however, provided that the convergence tolerance is small enough, analyst is assured that his/her solution is within proper material response and equilibrium. Solutions without enforced equilibrium are faster, and if they are done using explicit solvers, there is a requirement of small time step, which can then slow down the solution process. 6.3 Sub-structuring methods (Pecker and Johnson) Sub-Structuring Methods, Principles and Numerical Implementation (Pecker) NOTE: THIS is where Alain s section 6.3 is to be merged! Soil Structure Interaction CLASSI: A Linear Continuum Mechanics Approach (Johnson) NOTE This is where Jim s section 6.3 CLASSI is to be merged Discrete methods (finite elements and finite difference) 14

15 6.3.4 Foundation input motion (Reference to section 4.5 and 5.1) 6.4 SSI computational models (Jeremic and Pecker) Introduction Soil structure interaction computational models are developed with a focus on three components of the problem: Earthquake input motions, encompassing development of 1D or 2D or 3D motions, and their effective input in the SSI model, Soil/rock adjacent to structural foundations, with important geological (deep) and site (shallow) conditions near structure, contact zone between foundations and the soil/rock, and Structure, including structural foundations, embedded walls, and the superstructure It is advisable to develop models that will provide enough detail and accuracy to be able to address all the important issues. For example, for modeling higher frequencies of earthquake motions, analyst needs to develop finite element mesh that will be capable to propagate those frequencies and to document influence of numerical/mesh induced dissipation/damping of frequencies Soil/Rock Linear and Nonlinear Modelling Effective and Total Stress Analysis Soil and rock adjacent to structural foundations can be either dry or fully (or partially) saturated (Zienkiewicz et al., 1990; Lu and Likos, 2004). Dry Soil. In the case of dry soil, without pore fluid pressures, it is appropriate to use models that are only dependent on single phase stress, that is, a stress that is obtained from applying all the loads (static and/or dynamic) without any consideration of pore fluid pressures. Partially Saturated Soil. For partially saturated soil, effective stress principle (see equation 6.4 below) must also the include influence of gas (air) present in pore of soils. There are a number of different 15

16 methods to do that (Zienkiewicz et al., 1999; Lu and Likos, 2004), however computational frameworks that incorporate those methods are not yet well developed. Main approaches to modeling of soil behavior within a partially saturated zone of soil (a zone where water rises due to capillary effects) are dependent on two main types of partial saturation Voids of soil fully saturated with fluid mixed with air bubbles, water in pores is fully connected and can move and pressure in the mixture of water and air can propagate, with reduced bulk stiffness of water-air mixture. This type of partial saturation can be modeled using fully saturated approaches, given in section below. It is noted that bulk modulus of fluid-air mixture is (much) lower that that of fluid alone, and to be tested for. Therefor, only methods that assume fluid to be compressible should be used (u p U, u U, see section for details). In addition, permeability will change from a case of just fluid seeping through the soil, and additional testing for permeability of water-air mixture is warranted. It is also noted, that since this partial saturation is usually found above water table, (capillary rise), hydrostatic pore pressure can be suction. Voids of soil are full of air, with water covering thin contact zone between particles, creating water menisci, and contributing to the apparent cohesion of cohesionless soil material (think of wet sand at the beach, there is an apparent cohesion, until sand dries up). This type of partial saturation can be modeled using dry (unsaturated) modeling, where elasticplastic material models used are extended to include additional cohesion, that arises from thin water menisci connecting soil particles. Saturated Soil. In the case of full saturate, effective stress principle (Terzaghi et al., 1996) has to be applied. This is essential as for porous material (soil, rock, and sometimes concrete) mechanical behavior is controlled by the effective stresses. Effective stress is obtained from total stress acting on material (σ ij ), with reductions due to the pore fluid pressure: σ ij = σ ij δ ij p (6.4) where σ ij is effective stress tensor, σ ij is total stress tensor, δ ij is Kronecker delta (a diagonal matrix with numbers 1 on a diagonal and numbers 0 on non-diagonal positions, that is δ ij = 1, when i=j, and δ ij = 0, when i j), and p is the pore fluid pressure. We use standard mechanics of materials convention that tensile components of stress are positive, and so the pore fluid pressure p is negative 16

17 when in compression (Zienkiewicz et al., 1999). All the mechanical behavior of soils and rock is a function of the effective stress σ ij, which is affected by a full coupling with the pore fluid, through a pore fluid pressure p. A Note on Clays. Clay particles (platelets) are so small that their interaction with water is quite different from silt, sand and gravel. Clays feature chemically bonded water layer that surrounds clay platelets. Such water does not move freely and stays connected to clay platelets under working loads. Usually, clays are modeled as fully saturated soil material. In addition, clays feature very small permeability, so that, while the effective stress principle (from Equation 6.4) applies, pore fluid pressure does not change during fast (earthquake) loading. Hence clays should be analyzed using total stress stress analysis, where the initial total stress is a stress that is obtained from an effective stress calculation that takes into account hydrostatic pore fluid pressure. In other words, slays are modeled using undrained, total stress analysis, using effective stress (total stress reduced by the pore fluid pressure) for initializing total stress at the beginning of loading. Drained and Undrained Modeling Depending on the permeability of the soil, on relative rate of loading and seepage, and on boundary conditions (Atkinson, 1993), a decision needs to be made if analysis will be performed using drained or undrained behavior. Permeability of soil (k) can range from k > 10 2 m/s for gravel, 10 2 m/s > k > 10 5 m/s for sand, 10 5 m/s > k > 10 8 m/s for silt, to k < 10 8 m/s for clay. If we assume a unit hydraulic gradient (reduction of pore fluid pressure/head of 1m over the seepage path length of 1m), then for a dynamic loading of seconds (earthquake), and for a semi-permeable silt with k = 10 6 m/s, water can travel few millimeters. However, pore fluid pressure will propagate (much) faster (further) and will affect mechanical behavior of soil skeleton. This is due to high bulk modulus of water (K w = kn/m 2 ), which results in high speed of pressure waves in saturated soils. Thus a simple rule is that for earthquake loading, for gravel, sand and permeable silt, relative rate of loading and seepage requires use of drained analysis. For clays, and impermeable silt, it might be appropriate to use undrained analysis for such short loading. Drained Analysis Drained analysis is performed when permeability of soil, rate of loading and seepage, and boundary conditions allow for full movement of pore fluid and pore fluid pressures during loading event. As noted above, use of the effective stress σ ij for the analysis is essential, as is modeling of full coupling of pore fluid pressure with the mechanical behavior of soil skeleton. This is usually done 17

18 using theory of mixtures (Green and Naghdi, 1965; Eringen and Ingram, 1965; Ingram and Eringen, 1967; Zienkiewicz and Shiomi, 1984; Zienkiewicz et al., 1999) and will be elaborated upon in some detail in section During loading events, pore fluid pressures will dynamically change (pore fluid and pore fluid pressures will displace) and will affect the soil skeleton, through effective stress principle. All nonlinear (inelastic) material modeling applies to the effective stresses (σ ij ). Appropriate inelastic material models that are used for modeling of soil (as noted in section 6.4.2) should be used. Undrained Analysis Undrained analysis is performed when permeability of soil, rate of loading and seepage, and boundary conditions do not allow movement of pore fluid and pore fluid pressures during loading event. This is usually the case for clays and for low permeability silt. There are three main approaches to undrained analysis: Total stress approach, where there is no generation of excess pore fluid pressure (pore fluid pressure in addition to the hydraulic pressure), and soil is practically impermeable (clays and low permeability silt). In this case hydrostatic pore fluid (water) pressures are calculated prior to analysis, and effective stress is established for the soil. This approach assumes no change in pore fluid pressure. This usually happens for clays and low permeability silt, and due to very low permeability of such soils, a total stress analysis is warranted, using initial stress that is calculate based on an effective stress principle and known hydrostatic pore fluid pressure. Since pore fluid pressure does not affect shear strength (Muir Wood, 1990), for very low permeability soils (impermeable for all practical purposes), it is convenient to perform elastic-plastic analysis using undrained shear strength (c u ) within a total strain setup. Since only shear strength is used, and all the change in mean stress is taken by the pore fluid, material models using von Mises yield criteria can be used. Locally undrained analysis where excess pore fluid pressure (change from hydrostatic pore pressure) can be created. Excess pore fluid pressures can be created, due to compression effects on low permeability soil (usually silt). On the other hand, pore fluid suction can also be created due to dilatancy effects within granular material (silt). Due to very low permeability, pore fluid and pore fluid pressure does not move during loading, and hence, effective stress will change, and will affect constitutive behavior of soil. Analysis is essentially undrained, however, pore fluid pressure can and will change locally due to compression or dilatancy effects in granular soil. Appropriate inelastic (elastic-plastic) material models that are used for modeling of soil (as noted in section 6.4.2) should be used, while constitutive integration should take into account local undrained effects and 18

19 convert any change in voids into excess pore fluid pressure change (excess pore pressure). Very low permeability soils, that can, but to not have to develop excess pore fluid pressure can also be analyzed as fully drained continuum, while using very low, realistic permeability. In this case, although analysis is officially drained analysis, results will be very similar if not the same as for undrained behavior (one of two approaches above) due to use of very low, realistic permeability. Effective stress analysis is used, with explicit modeling of pore fluid pressure and a potential for pore fluid to displace and pore fluid pressure to move. However, due to very low permeability, and fast application of load (earthquake) no fluid will displace and no pore fluid pressure will propagate. This approach can be used for both cases noted above (total stress approach and locally undrained approach). While this approach is actually explicitly allowing for modeling of pore fluid movement, results for pore fluid displacement should show no movement. In that sense, this approach is modeling more variables than needed, as some results are known before simulations (there will be no movement of water nor pore fluid pressure). However, this approach can be used to verify modeling using the first two undrained approaches, as it is more general. It is noted that globally undrained problems, where for example soil is permeable, but boundary conditions prevent water from moving, should be treated as drained problems, while appropriate boundary conditions should prevent water from moving across impermeable boundaries. Linear and Nonlinear Elastic Models Linear and nonlinear elastic models are used for soil, rock and structural components. Linear elastic model that are used are usually isotropic, and are controlled by two constants, the Young s modulus E and the Poisson s ratio ν, or alternatively by the shear modulus G and the bulk modulus K. Nonlinear elastic models are used mostly in soil mechanics, There are a number of models proposed over years, tend to produce initial stiffness of a soil for given confinement of over-consolidation ratio (OCR) (Janbu, 1963; Duncan and Chang, 1970; Hardin, 1978; Lade and Nelson, 1987; Lade, 1988). Anisotropic material models are mostly used for modeling of usually anisotropic rock material (Amadei and Goodman, 1982; Amadei, 1983). Equivalent Linear Elastic Models Equivalent elastic models are linear elastic models where the elastic constants were determined from nonlinear elastic models, for a fixed shear strain value. They are secant stiffness 1D models and usually give relationship between shear stress (τ = σ xz ) and shear 19

20 strain (γ = 2ɛ xz ). Determination of secant shear stiffness is done iteratively, by performing 1D wave propagation simulations, and recording average high estimated strain (65% of maximum strain) for each level/depth. Such representative shear strain is then used to determine reduction of stiffness using modulus reduction curves (G/G max and the analysis is re-run. Stable secant stiffness values are usually reached after few iterations, typically 5-8. It is important to emphasize that equivalent elastic modeling is still essentially linear elastic modeling, with changed stiffness. More details are available in sections 3.2 and 4.5. Elastic-Plastic Models Elastic plastic modeling can be used in 1D, 2D and full 3D. A number of material models have been developed over years for both monotonic and cyclic modeling of materials. Material models for soil (Manzari and Dafalias, 1997; Taiebat and Dafalias, 2008; Papadimitriou et al., 2001; Dafalias et al., 2006; Lade, 1990; Pestana and Whittle, 1995; Prevost and Popescu, 1996; Mroz and Norris, 1982), rock (Lade and Kim, 1995; Hoek et al., 2002; Vorobiev, 2008) have been developed over last many years. It should be noted that 3D elastic plastic modeling is the most general approach to material modeling of soils and rock. If proper models are used (see section 6.2.1) it is possible to achieve modeling that is done using simplified modeling approaches described above (linear elastic, equivalent linear elastic, modulus reduction curves, etc.). However, calibration of models that can achieve such modeling sophistication requires expertise. The payoff is that important material response effects, that are usually neglected if simplified models are used, can be taken into account and properly modeled. As an example, soil volume change during shearing is a first order effects, however it is not taken into account if modulus reduction curves are used. A Note on Constitutive Level and Global Level Equilibrium. for constitutive integrations: There are two main types of algorithms Explicit or Forward Euler, is an algorithm that produces tangent stiffness tensor on the constitutive level. This algorithm does not enforce equilibrium and error in constitutive integrations (drift from the yield surface) is accumulated. This algorithm is simpler and faster than the implicit algorithm (next item) and is implemented and used in most (all) computer programs. Implicit or Backward Euler) is an algorithm that produces algorithmic (consistent) stiffness tensor (matrix) that can produce very fast convergence (quadratic for Newton scheme) on the global, 20

21 finite element equilibrium iterations. This algorithm is iterative and does enforce equilibrium (within user specified tolerance). It is usually slower than the explicit algorithm (see above) and implementation can be quite complicated, particularly for elastic plastic material models for soil and concrete (Crisfield, 1987; Jeremić and Sture, 1997; Jeremić, 2001). On the global, finite element level, there are two ways to advance the solutions Solution advancement without enforcing the equilibrium. In this case, solutions is produced using current tangent stiffness matrix (relying on the tangent stiffness tensor, developed on the constitutive level, as noted above). For each step of loading (static or dynamic) difference between applied loads and internal loads (stresses) is not checked for. This means that error in unbalanced forces is accumulating as computations progress. Usual remedy is to make steps small enough so that error is also reduced. However this reduction in step size (or time step size) can significantly increase computational times. In one specific instance, if lumped mass matrix is used, instead of a consistent mass matrix (which is theoretically more accurate), solution of a large system of equations can be completely circumvented. For particular explicit dynamic computations, only inverse of a diagonal mass matrix is required, which is trivial to obtain. Solution advancement with enforcement of the equilibrium. In this case, equilibrium is explicitly checked for, and if unbalanced forces are not balanced within certain (user specified) tolerance, an iterative scheme is used until equilibrium is achieved (within tolerance). Alternative method for ending iterations (instead of achieving equilibrium) is to check for iterative displacements and place a low limit below which iterations are not worth while any more and therefor end them. The most commonly used iteration methods are based on Newton iterative scheme Crisfield (1984). This approach is computationally demanding, however it does benefit the solution as it yields (close to) equilibrium solutions. In addition, if consistent (algorithmic) stiffness is used on the constitutive level (see Implicit constitutive algorithm above), a fast convergence (sometimes even close to quadratic) is achieved. Concluding note for both constitutive and global level solution advancement is that simpler methods (explicit, no equilibrium check) will lead to accumulating error (unbalance stress and force) and will thus render solutions that are not in equilibrium and are possibly quite wrong. This can be remedied by reducing step size (time increment), however computational times are then becoming long On the other 21

22 hand, methods that enforce equilibrium (within tolerance) are (much) more complicated to develop, implement and execute, yet they enforce equilibrium (again within tolerance) Structural models, linear and nonlinear: shells, plates, walls, beams, trusses, solids Linear and nonlinear structural models are not used as much in the industry for modeling and simulation of behavior of nuclear power plants (NPP). One of the main reasons is that NPPs are required to remain, effectively elastic during earthquakes. Nevertheless modeling of nonlinear effects in structures remains a viable proposition. It is important to note that inelastic behavior of structural components (trusses, beams, walls, plates, shells) features a localization of deformation (Rudnicki and Rice, 1975). While localization of deformation is also present in soils, soils are more ductile medium (unless they are very dense) and so inelastic treatment of deformation in soils, with possible localization, is more benign than treatment of localization of deformation in brittle concrete. Significant work has been done in modeling of nonlinear effects in mass concrete and concrete beams, plates, walls and shells (Feenstra, 1993; Feenstra and de Borst, 1995; de Borst and Feenstra, 1990; de Borst, 1987, 1986; de Borst, 1987; de Borst et al., 1993; Bićanić et al., 1993; Kang and Willam, 1996; Rizzi et al., 1996; Menetrey and Willam, 1995; Carol and Willam, 1997; Willam, 1989; Willam and Warnke, 1974; Etse and Willam, 1993; Scott et al., 2004, 2008; Spacone et al., 1996a,b; Scott and Fenves, 2006). The main issue is still that concrete structural elements still develop plastic hinges (localized deformation zones). Finite element results with localized deformation are known to be mesh dependent (change of mesh will change the result), and as such are hard to verify. Recent work on rectifuing this problem (Larsson and Runesson, 1993) shows promises, however these methods are still not widely accepted. One possible, rather successful solution relies on classical developments of Cosserat continua (Cosserat, 1909), where results looked very promising (Dietsche and Willam, 1992), however sophistication required by such analysis and lack of programs makes this approach still very exotic Contact Modeling In all soil-structure systems, there exist interfaces between structural foundations and the soil or rock beneath. There are two main modes of behavior of these interfaces, contacts: Normal contact where foundation and the soil/rock beneath interact in a normal stress mode. This mode of interaction comprises normal compressive stress, however it can also comprise gap 22

23 opening, as it is assumed that contact zone has zero tensile strength. Shear contact where foundation and the soil/rock beneath can develop frictional slip. Contact description provided here is based on recent work by Jeremić (2016) and Jeremić et al. ( ). Modeling of contact is done using contact finite elements. Simplest contact elements are based on a two node elements, the so called joint elements which were initially developed for modeling of rock joints. Typically normal and tangential stiffness were used to model the pressure and friction at the interface (Wriggers, 2002; Haraldsson and Wriggers, 2000; Desai and Siriwardane, 1984). The study of two dimensional and axisymmetric benchmark examples have been done by Olukoko et al. (1993) for linear elastic and isotropic contact problems. Study was done considering Coulomb s law for frictional behavior at the interface. In many cases the interaction of soil and structure is involved with frictional sliding of the contact surfaces, separation, and re-closure of the surfaces. These cases depend on the loading procedure and frictional parameters. Wriggers (2002) discussed how frictional contact is important for structural foundations under loading, pile foundations, soil anchors, and retaining walls. Two-dimensional frictional polynomial to segment contact elements are developed by Haraldsson and Wriggers (2000) based on non-associated frictional law and elastic-plastic tangential slip decomposition. Several benchmarks are presented by Konter (2005) in order to verify the the results of the finite element analyses performed on 2D and 3D modelings. In all proposed benchmarks the results were approximated pretty well with a 2D or an axisymmetric solutions. In addition, 3D analyses were performed and the results were compared with the 2D solutions. Contact Modeling Formulation The formulation for contact is represented by a discretization which establishes constraint equations and contact interface constitutive equations on a purely nodal basis. Such a formulation is called node-tonode contact (Wriggers, 2002). The variables adopted to formulate the model are shown in Figure 6.1: the force (F) and displacement vectors (u). Each vector is composed of three terms: the first one acts along the longitudinal direction (n local ) whereas the other two components lie on the orthogonal plane (m local and l local ). The total relative displacement additively decomposed into elastic and plastic components (6.6). F = [p ; t] T ; u = [v; g s ] T (6.5) 23